Introduction to Number Theory 0366-2140-01, 0366-2140-02.
Mikhail Borovoi
Fall 2018
Schedule:
Monday 10-12
Wednesday 11-13
Syllabus:
The course is an introductory course in basic number theory.
It assumes very little background. The topics include
- The Euclidean algorithm, greatest common divisor, unique factorisation
into primes, linear Diophantine equations
- Congruences, the Chinese Remainder Theorem
- The multiplicative group of reduced residue classes modulo n,
Fermat's Little Theorem
- The Euclidean algorithm for polynomials
- Primitive roots
- Quadratic congruences, Legendre's symbol and quadratic reciprocity,
Jacobi's symbol
- The Prime Number Theorem (without proof) and its applications
- Public Key Cryptography (RSA)
- Primality testing
- Arithmetic in the ring of Gaussian integers, sums of two squares,
Euclidean rings
- Rational points on curves of degree tqo
- Pythagorean triples and Fermat's Last Theorem
- Hensel's lemma
Bibliography
Any introductory book on number theory will be useful. For example, see:
- Elementary Number Theory, by D. Burton (available in Hebrew, published by the Open University).
- Elementary Number Theory: Primes, Congruences and Secrets, by William Stein (see online version).
- A more advanced text is "A Classical Introduction to Modern Number Theory"
by Ireland and Rosen.
From Lecture 1:
Non-unique factorization
Borovoi's lectures in Hebrew in 2013:
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
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21,
Example of an exam
Homework assignments:
#1,
#2,
#3,
#4,
#5,
#6,
A useful Internet resource: Mathematics - Stack Exchange
Contact me at: borovoi@post.tau.ac.il. Please write in Subject: Number Theory (in English).
Course homepage:
http://www.tau.ac.il/~borovoi/courses/NumberTheory/NT.html